If A = log base 32 of 1875 and B = log base 243 of 2187, then which of the following correctly describes the relationship between A and B?

Introduction / Context:
This question compares two logarithmic expressions with different bases and arguments. The aim is not to compute exact decimal values but to determine which of A or B is larger. Such comparisons are common in aptitude tests because they require factorization skills and a good understanding of how logarithms behave with respect to their bases and arguments.

Given Data / Assumptions:
A = log base 32 of 1875, written as log_32 1875.B = log base 243 of 2187, written as log_243 2187.All numbers involved are positive, and bases are not equal to 1.

Concept / Approach:
The key idea is to express all numbers in terms of their prime factorization, preferably using the same prime. For expression B, both 243 and 2187 are powers of 3, which allows an exact simplification. For expression A, while 32 is a power of 2, 1875 can be factorized to show its structure, and we can approximate its value or compare using change of base and inequalities. Once both expressions are in a manageable form, we can decide whether A is greater than, equal to or less than B.

Step-by-Step Solution:
Step 1: Factorize 243 and 2187. We have 243 = 3^5 and 2187 = 3^7.Step 2: Write B in terms of base 3: B = log_(3^5) (3^7).Step 3: By the exponent rule for logarithms, log_(a^k) (a^m) = m / k, so B = 7 / 5 = 1.4.Step 4: For A, note that 32 = 2^5 and factorize 1875 as 3 * 5^4.Step 5: Use the change of base formula: A = log 1875 / log 32 (any consistent log base).Step 6: Numerically, 1875 is much larger than 32, so the logarithm to base 32 of 1875 should be greater than 1. Indeed, calculation gives approximately A ≈ 2.17.Step 7: Compare the approximate values: A ≈ 2.17 while B = 1.4.Step 8: Therefore A is greater than B, so A > B.

Verification / Alternative check:
We can do a rough check without full decimal computation. Since 32^1 = 32 and 32^2 = 1024, while 32^3 = 32768, the number 1875 lies between 32^2 and 32^3. That means A is between 2 and 3. By contrast, for base 243, we have 243^1 = 243 and 243^2 = 59049, with 2187 lying between 243^1 and 243^2, but much closer to 243 than to 59049. Therefore B is clearly a number between 1 and 2 and closer to 1, which supports the conclusion that A > B.

Why Other Options Are Wrong:
The option A < B contradicts the approximate numerical comparison. The option A = B would require both expressions to have exactly the same value, which is impossible since B simplifies to 7 / 5 while A is clearly larger than 2. The option that the relationship cannot be determined ignores the clear factorization and comparison that have been carried out. Hence only A > B is correct.

Common Pitfalls:
Some learners try to convert everything to base 10 and then fall into rounding errors. Others mis-handle the property log_(a^k) (a^m) = m / k or factorize the numbers incorrectly. Another common mistake is to rely on intuition instead of performing a systematic comparison. Using prime factorization and simple inequalities avoids these issues.

Final Answer:
The correct relationship is A > B, so A is greater than B.